I have tried the Paraguay classifieds mentioned by Armand above. There are no rentals listed and there have been no replies to the want ad I posted several months ago.
Right now, I am staying in a hotel with my cat, waiting for my flight to Asuncion, which leaves in 2 days. We are in this situation because the owner of an Apart hotel in Asuncion promised me a room several months ago, then rented it out without bothering to notify me. I would've given them a deposit to hold the room, but I was told it wasn't necessary. It took 3 weeks to receive my PY visa. In that time, the owners rented my room to someone else. If I hadn't emailed them my arrival date, I wouldn't have known I had no place to live until I showed up on their doorstep. If they'd just bothered to tell me they were renting the room to someone else, I wouldn't be caught homeless just 2 days before my flight!
They advertise on this forum, so it's probably not appropriate to post their name. If you send me a PM, I'll tell you who it is so you and your friends can avoid getting screwed like me.